[viraghr]

Andrew, I perfectly see what you're saying, but I am telling you it is absolutely no problem that there's a false statement.

Remember that we are arguing from the false assumption that you start with the finite set of all primes, up to some largest prime, L. Call this false assumption A. It's fine to make false statements that follow from A, no problem at all. Indeed this is the point.

One definition of a prime is that there are no primes smaller than it in its prime factorization. We'll call this definition NSPIPF - no smaller prime in prime factorization. Is NSPIPF an OK test for primality? Sure.

So when you get to new_number, produce the prime factorization. Does it meet the definition of NSPIPF? Yes, because we made it relatively prime to every prime (under assumption A).

It passes primality test NSPIPF. Under A. And therefore is prime. Under A. (This is the part you and others object to, but it's absolutely flawless application of the NSPIPF primality test.) It doesn't matter that in some other way I could also get to a contradiction. For now this is what we do.

Under A, using NSPIPF primality test, we just proved new_number is prime.

Next we show that this new_number definitely wasn't in the set of all primes, since it's larger than L, having had L and a bunch of other positive integers as factors and then one added to that for good measure. It is at this point that we show explicitly that A cannot be true, because we used A to produce a prime that wasn't in the set of all primes.

It doesn't matter if that was a false statement!

I think all this is in my original comment and it is a flawless proof by contradiction. I asked you to "Suppose there are just finite primes, up to some largest" and you gave up when you started seeing false statements, rather than at the end of the paragraph where I presented the conclusion in black and white.

[andrewla]

Nobody is contesting that the conclusion is true, just that the proof is unsound as written.

Here you've introduced a definition of prime that is different than the usual one, and is in fact self-recursive. NSPIPF is "no smaller prime in prime factorization". What is the first "prime" in this definition? Is it a number such that there is "no smaller prime in prime factorization"? What is the second "prime" in that definition? Is that the usual "prime factorization", or is it an NSPIPF factorization? In other words, how would you prove that 2 is prime given the NSPIPF definition?

It is more natural to talk about primality as a test that can be done independent of any assumptions about other primes, but rather as a matter of whether it can be expressed as the multiple of some number other than 1 and itself. That way we can make a precise statement about the product of the finite set of primes plus one without reference to the set of primes.

[viraghr]

>It is more natural to talk about primality as a test that can be done independent of any assumptions about other primes, but rather as a matter of whether it can be expressed as the multiple of some number other than 1 and itself

This is clearly not what I was doing. I clearly referred to having no smaller prime factors. Anyway this aside is tiresome, it's like poking me for saying "every positive integer has a prime factorization" and then asking, okay, so what about 1 or something. I think my proof is fine and I'm not going to defend it anymore.